A Bayesian Approach for Defect Arrival Modelling and Inspection Planning for Linear Assets Using Negative Binomial Process

Developing practical inspection policies is essential for managing linear assets, yet comprehensive inspection data are often limited relative to the network's spatial extent. Inspections are typically conducted over long timeframes, with records capturing only partial segments of the asset in a stochastic manner both in timing and extent. This study addresses defect arrival modelling and inspection planning under such inspection regimes. A Negative Binomial Process (NBP) is proposed to model the over-dispersion observed in defect records resulting from inspection variability. A hierarchical Bayesian framework is employed to estimate model parameters, balancing bias and variance in the presence of limited data. Posterior distributions are derived using Markov Chain Monte Carlo (MCMC) methods. Decision support for inspection planning is formulated from the estimated model, using the expected age of the first defect occurrence as a proxy for risk. The application of the developed methodology is examined through a real-world case study of surface defect arrivals on a rail network and benchmarked against a more traditional Poisson process. The results demonstrate that the NBP model offers a more accurate characterization of defect arrival uncertainty, thereby enabling a more reliable assessment of the risk associated with a given inspection regime. The proposed method provides decision makers with a tool to assess the impact of partial inspections on risk for partially inspected linear assets.